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# Normed Vector Space

Module by: Phil Schniter. E-mail the author

Summary: This module introduces normed vector space.

Now we equip a vector space VV with a notion of "size".

• A norm is a function ( :VR : V ) such that the following properties hold ( x , y ,xVyV: xVyV x y x V y V and α ,α: α α α ):
1. x0 x 0 with equality iff x=0 x 0
2. αx=|α|x α x α x
3. x+yx+y x y x y , (the triangle inequality).
In simple terms, the norm measures the size of a vector. Adding the norm operation to a vector space yields a normed vector space. Important example include:
1. V=RN V N , x 0 x N - 1 T i =0N1 x i 2xTx x 0 x N - 1 i 0 N 1 x i 2 x x
2. V=CN V N , x 0 x N - 1 T i =0N1| x i |2xHx x 0 x N - 1 i 0 N 1 x i 2 x x
3. V= l p V l p , xnn=|xn|p1p x n n x n p 1 p
4. V= p V p , ft|ft|pdt 1p f t t f t p 1 p

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